Writing and Using Inequalities
Learning Objective(s)
· Develop inequalities to represent real world situations and use them to solve problems.
Introduction
Inequalities are used all the time in the world around us—we just have to know where to look. Figuring out how to interpret the language of inequalities is an important step toward learning how to solve them in everyday contexts.
You are confronted with mathematical inequalities almost every day, but you may not notice them because they are so familiar. Think about the following situations: speed limits on the highway, minimum payments on credit card bills, number of text messages you can send each month from your cell phone, and the amount of time it will take to get from home to school. All of these can be represented as mathematical inequalities. And, in fact, you use mathematical thinking as you consider these situations on a daytoday basis.
Situation  Mathematical Inequality 
Speed limit  Legal speed on the highway ≤ 65 miles per hour 
Credit card  Monthly payment ≥ 10% of your balance in that billing cycle 
Text messaging  Allowable number of text messages per month ≤ 250 
Travel time  Time needed to walk from home to school ≥ 18 minutes 
When we talk about these situations, we often refer to limits, such as “the speed limit is 65 miles per hour” or “I have a limit of 250 text messages per month.” However, we don’t have to travel at exactly 65 miles per hour on the highway, or send and receive precisely 250 test messages per month—the limit only establishes a boundary for what is allowable. Thinking about these situations as inequalities provides a fuller picture of what is possible.
Consider this problem:
An 18wheel truck stops at a weigh station before passing over a bridge. The weight limit on the bridge is 65,000 pounds. The cab (front) of the truck weighs 20,000 pounds, and the trailer (back) of the truck weighs 12,000 pounds when empty. In pounds, how much cargo can the truck carry and still be allowed to cross the bridge?
This problem offers us an upper limit—65,000 pounds—but we are interested in finding out the full range of possibilities for the weight of the truck’s cargo. We can represent the situation using the following inequality, where c is the weight (in pounds) of the truck’s cargo:
cab weight  +  trailer weight  +  cargo weight  ≤  allowable weight 
20,000  +  12,000  +  c  ≤  65,000 
Solving this inequality for c, we find that c ≤ 33,000. This means that the weight of the cargo in the truck can be anywhere between 0 pounds and 33,000 pounds and the truck will be allowed to cross the bridge.
20,000 + 12,000 + c  ≤  65,000 
20,000 + 12,000 + c – 32,000  ≤  65,000 – 32,000 
c  ≤  33,000 
Understanding Context
When you are solving or building these inequalities, it is important to know which inequality symbol you should use. Watch for certain phrases that will tip you off:
Phrase  Inequality 
“a is more than b”  a > b 
“a is at least b”  a ≥ b 
“a is less than b”  a < b 
“a is at most b;” or  a ≤ b 
Many problems, though, will not explicitly use words like “at least” or “is less than.” So how do you figure out which symbol is appropriate in a given situation?
The key is to think about the context of the problem, and to relate the context to one of the situations listed in the table. Context refers to the reallife situation is which the problem takes place.
So, for example, think back to the truck problem that we solved above. The maximum weight allowed on this bridge was 65,000 pounds. We can also think of this relationship using the language of inequalities used in the table: the total weight of the cab, trailer, and cargo had to be no more than 65,000. Once we have identified the relationship between the two quantities we can put in the appropriate symbol.
Consider another problem that requires us to look at context:
Erykah has found three pairs of running sneakers that she likes, costing $150, $159, and $179. She has saved $31 already, and she has a job where she earns $8.50 per hour. How many hours will she have to work in order to afford any of these sneakers?
Notice that this problem is not asking us to find out the number of hours Erykah will have to work to afford a specific sneaker—it is asking about any of these sneakers. Since the least expensive sneaker is $150, we need to create an inequality that shows how much she needs to work in order to have “at least $150”—or, in math symbols, “≥ $150”. We will use the variable h to represent the number of hours she works.
Hourly wage  •  Number of hours she works  +  Money she has saved  ≥  Price of the least expensive sneakers 
($8.50  •  h)  +  $31  ≥  $150.00 
We can solve this inequality like this:
Example  
Problem  8.5h + 31  ≥  150 
 8.5h + 31 31  ≥  150 – 31 
 8.5h  ≥  119 
 _{}
 ≥  _{} 
Answer  h  ≥  14

Erykah needs to work at least 14 hours to afford any of the running sneakers. If she works more hours, she’ll be able afford the more expensive models.
If s represents the number of books that Sam has, and g represents the number of books that Gerard has, which inequality models the situation “Sam has more than twice the number of books as Gerard?”
A) s > 2g
B) s < 2g
C) g < 2s
D) g > 2s
Absolute Value Inequalities
Some real life inequalities require the use of absolute value. Once again, there are certain phrases to look for in a description of these kinds of situations:
Phrase 
“margin of error” 
“more or less” 
“plus or minus” 
“give or take” 
As before, you’ll still need to consider the context of the problem to be sure if absolute value is appropriate. Here’s an example:
At a tannery, a worker uses a machine to cut large strips of leather down to a consistent size in order to make belts. The belts are supposed to be 35 inches long, but the worker is allowed a margin of error of up to _{} inch for the belt to be of acceptable length. Write an inequality that represents the range of acceptable belt lengths.
We know that the belts have to be 35 inches long, give or take up to _{} inch. How do we account for this margin of error? We use an absolute value inequality, like this:
Target length of belt    Actual length of belt  <  Acceptable margin of error 
35    b  <  _{} 
In this inequality, the term 35 – b represents the absolute value of the difference between the target length of the belt and the actual length of the belt. It does not matter if the belt is a little longer or a little shorter—either is acceptable. Note that the order of these terms could have been reversed, and we could have written b – 35. We care about distance, but not direction. We used the inequality symbol <, because the difference must be less than the margin of error.
Let’s solve the inequality:
Example  
 Problem 35 – b < _{} 

 
35 – b  <  _{} 
 (35 – b)  <  _{}  
35 – b – 35  <  _{}  35 
 35 + b  <  _{}  
b  <  _{} 
 35 + b + 35  <  _{} + 35  
1 • b  >  1 • _{} 
 Answer:  b  <  _{}  
Answer:  b  >  _{} 



 
Any value between (but not including) _{} and _{} will satisfy this inequality—this represents the acceptable range of values for the belt lengths.
Summary
Inequalities can be used to model a number of reallife situations. When converting such word problems into inequalities, begin by identifying how the quantities relate to each other, and then pick the inequality symbol that is appropriate for that situation. When solving these problems, remember that the solution will be a range of possibilities—inequalities do not have a single answer, as equations do. Absolute value inequalities can be used to model situations where margin of error is a concern.